Chapter 1: Picturing Distributions with Graphs1 Picturing Distributions with Graphs BPS chapter 1 © 2006 W. H. Freeman and Company

Chapter 1: Picturing Distributions with Graphs2 Objectives (BPS chapter 1) Picturing Distributions with Graphs u Individuals and variables u Two types of data: categorical and quantitative u Ways to chart quantitative data: histograms and stemplots u Interpreting histograms u Time plots

Chapter 1: Picturing Distributions with Graphs3 Individuals and variables Individuals are the objects described by a set of data. Individuals may be people, animals, or things. –Example: Freshmen, 6-week-old babies, golden retrievers, fields of corn, cells A variable is any characteristic of an individual. A variable can take different values for different individuals. –Example: Age, height, blood pressure, ethnicity, leaf length, first language

Chapter 1: Picturing Distributions with Graphs4 Two types of variables

Chapter 1: Picturing Distributions with Graphs5 Two types of variables Cont. u quantitative –Something that can be counted or measured for each individual and then added, subtracted, averaged, etc., across individuals in the population. –Example: How tall you are, your age, your blood cholesterol level, the number of credit cards you own. u categorical –Something that falls into one of several categories. What can be counted is the count or proportion of individuals in each category. –Example: Your blood type ( A, B, AB, O ), your hair color, your ethnicity, whether you paid income tax last tax year or not.

Chapter 1: Picturing Distributions with Graphs6 Two types of variables Cont. See page 5 for description of variables.

Chapter 1: Picturing Distributions with Graphs7 Exploratory Data Analysis (EDA) u Can be a table, graph, or function Distribution  Tells what values a variable takes and how often it takes these values.  Can be a table, graph, or function.

Chapter 1: Picturing Distributions with Graphs8 EDA Cont. The W’s: Who What (in what units) When Where Why u Can be a table, graph, or function

Chapter 1: Picturing Distributions with Graphs9 “University Students are Healthier than you Think” A Spring 2002 random undergraduate classroom survey of n=810 students was conducted by the Office of Health Promotion within a University Student Health Services, Division of Student Affairs. Statistics from this survey led to the following conclusions: - most students (67%) have 0-4 drinks when they go out - most (69%) have had 0-1 sex partners in the past year - most (76%) either don’t drink, or use designated drivers if they do Identify the W’s.

Chapter 1: Picturing Distributions with Graphs10 Important Characteristics of Data The following characteristics of data are usually important: Center: An average value that indicates where the middle of the dataset is located. Variation/Spread: A measure of the amount of variation in the data (average variation from the center). Distribution: The shape of the distribution of the data (symmetric, uniform or skewed). Outliers: Sample values that lie far away from the vast majority of the other values. Time: Trend- changing characteristics of the data over time.

Chapter 1: Picturing Distributions with Graphs11 Presentation of Data To better analyze a dataset, we first present it in a summarized form using: u Frequency Tables u Pictures or Graphs u Numerical Summaries (Center and Variation)

Chapter 1: Picturing Distributions with Graphs12 Frequency Tables A frequency table lists data values (either individually or by groups of intervals called classes ), along with the number of items that fall into each class (frequency). Example: Test ScoreFrequency This frequency table has 7 classes (0-4,5-9,10-14,15-19,20-24,25- 29,30-34). The frequency represents the number of students receiving that score.

Chapter 1: Picturing Distributions with Graphs13 Example The heights (in inches) of 30 students are as follows: Create a frequency table for the above data using the classes 60-61, 62-63, etc.

Chapter 1: Picturing Distributions with Graphs14 RELATIVE FREQUENCY TABLES Relative frequency = frequency / total # of items The relative frequency gives the percent of items in each class. A relative frequency table is a frequency table with a column for the relative frequencies. The relative frequencies might not add to 1 (100%) due to rounding. Example: Construct a relative frequency table for our last example.

Chapter 1: Picturing Distributions with Graphs15 Graphs for Categorical Data  A picture (a good one) is worth a thousand words.  Bar Graph  Horizontal axis represents the categories.  Vertical axis represents the frequencies.  A bar whose height is proportional to the frequency is drawn centered at the category.

Chapter 1: Picturing Distributions with Graphs16 Example The following table gives the grade distributions of a Math 161 Test: GradeFrequency A 5 B 7 C 12 D 5 F 3 Draw a bar graph for the data.

Chapter 1: Picturing Distributions with Graphs17 Bar graph sorted by rank  Easy to analyze Top 10 causes of death in the U.S., 2001 Sorted alphabetically  Much less useful

Chapter 1: Picturing Distributions with Graphs18 Graphs for Categorical Data  Double (Side-by-side) Bar Graphs Used to compare two different distributions.  For each category, draw two adjacent bars (one for each distribution).

Chapter 1: Picturing Distributions with Graphs19 Example Suppose we now have the grades for two sections of Math 161: Grade Frequency Section 1 Section 2 A 5 3 B 7 5 C 12 9 D 5 4 F 3 1 Draw a double bar graph for the data.

Chapter 1: Picturing Distributions with Graphs20 Example Are the frequency bar graphs the right way to compare the performance of the two sections? Note that the class sizes are not the same. What would be a better way to compare the two sections? Grade Frequency Section 1 Section 2 A 5 3 B 7 5 C 12 9 D 5 4 F 3 1

Chapter 1: Picturing Distributions with Graphs21 Graphs for Categorical Data  Pie Chart  Shows the whole group of categories in a circle.  Shows the parts of some whole.  The area of the sector representing a category is proportional to the frequency of the category.

Chapter 1: Picturing Distributions with Graphs22 Example The following table gives the grade distributions of a Math 161 Test: GradeFrequency A 5 B 7 C 12 D 5 F 3 Draw a pie chart for the data.

Chapter 1: Picturing Distributions with Graphs23 Pictographs A picture of a set of small figures or icons used to represent data, and often to represent trends. Usually, the icons are suggestively related to the data being represented. They can be misleading.

Chapter 1: Picturing Distributions with Graphs24 Pictographs Double the length, width, and height of a cube, and the volume increases by a factor of eight.

Chapter 1: Picturing Distributions with Graphs25

Chapter 1: Picturing Distributions with Graphs26 Time (Line) Graphs u A time graph shows behavior over time. u Time is always on the horizontal axis. u Look for an overall pattern (trend). u Look for patterns that repeat at known regular intervals (seasonal variations). u Look for any striking deviations that might indicate unusual occurrences.

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Chapter 1: Picturing Distributions with Graphs28

Chapter 1: Picturing Distributions with Graphs29 Misleading Graphs Changing the scale of a line graph or a bar graph can make increases or decreases appear more rapid. Both graphs plot the same data. Which one makes the increase in cancer deaths appear more rapid? Which graph would a cancer advocate use?

Chapter 1: Picturing Distributions with Graphs30 Bachelor High School Degree Diploma Salaries of People with Bachelor’s Degrees and with High School Diplomas Which graph is misleading? $40,000 30,000 25,000 20,000 $40,500 $24,400 35,000 $40,000 20,000 10,000 0 $40,500 $24,400 30,000 Bachelor High School Degree Diploma (a)(b)

Chapter 1: Picturing Distributions with Graphs31 Important skills uCuComputer programs will construct plots and calculate summary statistics automatically. uTuThe important skills for people are: –k–knowing what to use when. –I–Interpretation. uTuThe tools used to analyze and summarize data depend upon the type of variable one is interested in.

Chapter 1: Picturing Distributions with Graphs32 Principles for plots The way plots are used depends upon the purpose for which they are being used: u Exploration –Principle: Look at the data in as many different ways as possible searching for its important features. u Communication to others (follows exploration) –Principle : Be selective. Choose the displays that best show to a reader features you have observed.

Chapter 1: Picturing Distributions with Graphs33 Making Good Graphs u Title your graph. u Make sure labels and legends describe variables and their measurement units. Be careful with the scales used. u Make the data stand out. Avoid distracting grids, artwork, etc. u Pay attention to what the eye sees. Avoid pictograms and tacky effects.

Chapter 1: Picturing Distributions with Graphs34 Key Concepts u Categorical and Quantitative Variables u Distributions u Pie Charts u Bar Graphs u Line Graphs u Techniques for Making Good Graphs

Chapter 1: Picturing Distributions with Graphs35 Graphs for Quantitative Variables

Chapter 1: Picturing Distributions with Graphs36 Stemplots (Stem-and-Leaf Plots) u For quantitative variables. u Separate each observation into a stem (first part of the number) and a leaf (the remaining part of the number). u Usually, the last digit is used as the leaf and the remaining digits form the stem. u If using the last digits as they are results in a lot of stem values, we could round the numbers to more convenient values. u Write the stems in a vertical column; draw a vertical line to the right of the stems. u Write each leaf in the row to the right of its stem; order leaves if desired.

Chapter 1: Picturing Distributions with Graphs37 Weight Data Weights (in pounds) for a group of 40 students.

Chapter 1: Picturing Distributions with Graphs38 Weight Data: Stemplot (Stem and Leaf Plot) Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s

Chapter 1: Picturing Distributions with Graphs39 Stem-and-Leaf Plots Double stemmed (expanded) stem-and-leaf: u If there are a lot of leaves on one stem, we could break it up into two stems one for the digits 0-4 and the other for the digits 5-9. u Back to back : Used to compare two different sets of data.

Chapter 1: Picturing Distributions with Graphs40 Histogram A histogram is a bar graph in which u the horizontal axis represents the items or classes. u the vertical axis represents the frequencies. u the height of the bars are proportional to the frequencies. u There are usually no gaps between the bars (unless some classes have 0 frequencies). u To draw a histogram, we first need to construct a frequency table. u Example: draw a histogram for our weights example. u The number of classes can affect the shape of the histogram.

Chapter 1: Picturing Distributions with Graphs41 Not summarized enough Too summarized Same data set

Chapter 1: Picturing Distributions with Graphs42 Weight Data: Frequency Table * Left endpoint is included in the group, right endpoint is not.

Chapter 1: Picturing Distributions with Graphs43 Weight Data: Histogram * Left endpoint is included in the group, right endpoint is not.

Chapter 1: Picturing Distributions with Graphs44

Chapter 1: Picturing Distributions with Graphs45 Shape of the Data u Symmetric –bell-shaped –other symmetric shapes u Asymmetric –skewed to the right –skewed to the left u Unimodal, bimodal

Chapter 1: Picturing Distributions with Graphs46 Most common distribution shapes u A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution  Not all distributions have a simple overall shape, especially when there are few observations. Right Skewed distribution  A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.

Chapter 1: Picturing Distributions with Graphs47 Symmetric Distributions Bell-ShapedMound-Shaped

Chapter 1: Picturing Distributions with Graphs48 Symmetric Distributions Uniform

Chapter 1: Picturing Distributions with Graphs49 Asymmetric Distributions Skewed to the LeftSkewed to the Right

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Chapter 1: Picturing Distributions with Graphs51 Number of Books Read for Pleasure

Chapter 1: Picturing Distributions with Graphs52 Outliers u Extreme values, far from the rest of the data. u May occur naturally. u May occur due to error in recording. u May occur due to error in measuring. u Observational unit may be fundamentally different.

Chapter 1: Picturing Distributions with Graphs53 Key Concepts u Displays (Stemplots & Histograms) u Graph Shapes –Symmetric –Skewed to the Right –Skewed to the Left u Outliers